Criticality of colloids with three distinct interaction patches: As simple as $A,B,C$?

We systematically study the phase behavior of a simple model of associating fluids which consists of hard spherical particles with three short-ranged attractive sites on their surfaces (sticky spots or patches), of types $A,B$, and $C$, that can form bonds with energy ${\epsilon }_{ij}$ $\left(i,j=A,B,C\right)$. We consider realizations of the model with one, two, or three nonzero ${\epsilon }_{ij}$. Using Wertheim's first order perturbation theory of association, we establish the minimum requirements on the bond energies for the model to exhibit a liquid-vapor critical point, and investigate the nature of criticality in each case. As a preliminary, we rigorously show that, within this theory, particles with $M$ identical sites do not condense if $M<3$, a result that was previously conjectured, but never proved.

Figure 1

(a) Patchy colloid with three patches, labeled $A,B$, and $C$, which may have different attraction ranges and strengths. (b) X junction: it is a bond (here a $BB$ bond) connecting two inner particles of two linear chains (here made up of $AC$ bonds). (c) Y junction: it is a bond (here an $AB$ bond) between an end particle of one linear chain and an inner particle of another linear chain (here the chains are made up of $AC$ bonds).

Figure 2

Critical point of $ABC$ patchy colloids with ${\epsilon }_{AA}\ne 0$ and ${\epsilon }_{BC}\ne 0$ vs ${\epsilon }_{AA}/{\epsilon }_{BC}$. (a) Critical density ${\rho }_c^{*}$; (b) critical temperature $T_c^{*}=k_B{T}_c/{\epsilon }_{BC}$; (c) $X_A,X_B$, and $X_C$, the fractions of unbonded $A,B$, and $C$ sites, respectively. $T_c^{*}\to 0$ as ${\epsilon }_{AA}/{\epsilon }_{BC}\to 0$, indicating X-junction-driven criticality (the dotted portion of the $T_c^{*}$ curve is a linear extrapolation, as this range is numerically inaccessible). Because the model is invariant under exchange $B\leftrightarrow C$, the curves for $X_B$ and $X_C$ are identical. Note the nonmonotonic behavior of ${\rho }_c^{*}$, which is typical of self-assembling systems [26].

Figure 3

Summary of criticality of $ABC$ patchy colloids with ${\epsilon }_{AA}\ne 0,{\epsilon }_{BB}\ne 0$, and ${\epsilon }_{AC}\ne 0$. Here, $r_B={\epsilon }_{BB}/{\epsilon }_{AA}$ and $r_C={\epsilon }_{AC}/{\epsilon }_{AA}$. The lines are analytical predictions (see Appendix pp4 for details), the stars are numerical results.

Figure 4

Summary of criticality of $ABC$ patchy colloids with ${\epsilon }_{AA}\ne 0,{\epsilon }_{BC}\ne 0$, and ${\epsilon }_{AC}\ne 0$. Here, $r_A={\epsilon }_{AA}/{\epsilon }_{AC}$ and $r_B={\epsilon }_{BC}/{\epsilon }_{AC}$. The lines are analytical predictions (see Appendix pp5 for details), the stars are numerical results.

Figure 5

Summary of criticality of $ABC$ patchy colloids with ${\epsilon }_{AB}\ne 0,{\epsilon }_{BC}\ne 0$, and ${\epsilon }_{AC}\ne 0$. Here, $r_A={\epsilon }_{AC}/{\epsilon }_{AB}$ and $r_B={\epsilon }_{BC}/{\epsilon }_{AB}$. The lines are analytical predictions (see Appendix pp6 for details), the stars are numerical results. Note the symmetry under exchange $A\leftrightarrow B$.

Figure 6

Critical point of $ABC$ patchy colloids with ${\epsilon }_{AA}\ne 0,{\epsilon }_{BC}\ne 0$, and ${\epsilon }_{AC}\ne 0$ vs ${\epsilon }_{AA}/{\epsilon }_{AC}$. (a) Critical density ${\rho }_c^{*}$; (b) critical temperature $T_c^{*}=k_B{T}_c/{\epsilon }_{AC}$; (c) $X_A,X_B$, and $X_C$, the fractions of unbonded $A,B$, and $C$ sites, respectively. [In part (b) the dotted portions of the curves are linear extrapolations, as this range is numerically inaccessible.] For ${\epsilon }_{BC}/{\epsilon }_{AC}=1.0$, there is X-junction-driven criticality: $T_c^{*}\to 0$ as ${\epsilon }_{AA}/{\epsilon }_{AC}\to 0$. In contrast, for ${\epsilon }_{BC}/{\epsilon }_{AC}=0.75$ there is no critical point if ${\epsilon }_{BC}/{\epsilon }_{AC}<0.25$, and Y-junction-driven criticality if ${\epsilon }_{BC}/{\epsilon }_{AC}>0.25$ (cf. Fig. 4); the boundary between the two regimes is marked by the dashed vertical line.